How to Solve the Non-linear Differential Equation in Radial Ink Diffusion?

In summary, the conversation discusses the Separation of Variables method and the resulting expressions for T(t) and R(r). It also addresses a multiple choice question about the dimensions of D in the diffusion equation, with R having dimensions of length and D being dimensionless. It is determined that R^2/D is the only answer with dimensions of time.
  • #1
tanaygupta2000
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Homework Statement
The concentration p(r,t) of ink diffusing in water is governed by the diffusion equation
∂p/∂t = D∇^2(p)
where D is a parameter known as diffusion constant. What is the average time taken for a molecule of ink to spread by a root mean square distance R?
(a) √(R/D)
(b) R/√D
(c) R^2 /D
(d) RD
Relevant Equations
∂p/∂t = D∇^2(p)
I assumed p(r,t) as p(r,t) = R(r)T(t) as Separation of Variables method. I got the expression of T(t) as
T(t) = C1eC2t

and got a non-linear differential equation in R(r) as
d2R/dr2 + (2/r)dR/dr - (C/D)R = 0

(I assumed r to be the radial distance in spherical coordinates)
Now I'm not getting how to solve this differential equation.
 
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  • #2
You don't need to solve a differential equation to answer this multiple choice question. Only one answer passes the test of dimensional analysis.
 
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  • #3
kuruman said:
You don't need to solve a differential equation to answer this multiple choice question. Only one answer passes the test of dimensional analysis.
R has the dimensions of length and D is a dimensionless quantity.
None of the options give the dimensions of time.
 
  • #4
D is not dimensionless. What are its dimensions as per the diffusion equation you posted?
 
  • #5
kuruman said:
D is not dimensionless. What are its dimensions as per the diffusion equation you posted?
I'm assuming p(r,t) = R(r)T(t).
By this assumption, ∂p/∂t has the dimensions of [LT]/[T] = [L]
and ∇2p(r,t) has the dimensions of [LT]/[L2] = [L-1T]

Hence only R2/D has the dimensions of time, hence the correct answer !
 
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  • #6
tanaygupta2000 said:
I'm assuming p(r,t) = R(r)T(t).
By this assumption, ∂p/∂t has the dimensions of [LT]/[T] = [L]
and ∇2p(r,t) has the dimensions of [LT]/[L2] = [L-1T]

Hence only R2/D has the dimensions of time, hence the correct answer !
Yes.
 
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Related to How to Solve the Non-linear Differential Equation in Radial Ink Diffusion?

1. How does diffusion of ink in water occur?

Diffusion of ink in water is the process by which the ink particles spread out evenly in the water due to random molecular motion. This occurs because the ink particles have a higher concentration in one area and a lower concentration in another, causing them to move from an area of high concentration to an area of low concentration until they are evenly distributed.

2. What factors affect the rate of diffusion of ink in water?

The rate of diffusion of ink in water can be affected by various factors such as temperature, concentration of ink, and size of the ink particles. Higher temperatures and higher concentrations of ink can increase the rate of diffusion, while larger ink particles can slow down the process.

3. How is diffusion of ink in water different from other types of diffusion?

Diffusion of ink in water is a specific type of diffusion known as molecular diffusion, where molecules or particles move from an area of high concentration to an area of low concentration. Other types of diffusion include thermal diffusion, where particles move due to differences in temperature, and forced diffusion, where particles are pushed by external forces.

4. What are some real-life applications of diffusion of ink in water?

Diffusion of ink in water is a commonly observed phenomenon in everyday life. It is used in chromatography, a technique used to separate and analyze mixtures, as well as in the production of dye and ink. It is also an important process in the functioning of biological systems, such as the diffusion of oxygen and nutrients in cells.

5. Can the diffusion of ink in water be reversed?

No, the diffusion of ink in water cannot be reversed. Once the ink particles have spread out evenly in the water, they will continue to move randomly and remain evenly distributed. This is due to the second law of thermodynamics, which states that systems tend to move towards a state of maximum entropy, where the energy is evenly distributed.

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